Optimisation Constraints

Transcription

1 Optimisation Constraints Nicolas Beldiceanu Mats Carlsson SICS Lägerhyddsvägen 5 SE Uppsala, Sweden {nicolas,matsc}@sics.se

2 Situation for LP/IP Optimisation is a basic idea in math programming: minimize cx subject to Ax b some x integer The underlying Simplex algorithm uses the cost function directly in its search for a minimal cx of the relaxed problem.

3 Situation for CP Traditionally a weak point, but in the recent years : Global constraints with cost variables : scheduling with different cost criteria (makespan,...), assignment with cost, relaxation with cost (alldifferent, disjunctive scheduling). Communication with linear programming, convex hull relaxation for different global constraints: element, cumulative.

4 Where to find the main references? Scheduling+optimization : Philippe Baptiste CP+LP : John Hooker Cost Filtering : Michela Milano

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6 Optimisation Left to right order, increasing values. labeling([costvar Rest]) First solution found is optimal.

7 Optimisation 2 Typically a labeling that fixes all variables (and also CostVar) minimize(goal, CostVar) Restart from scratch each time it finds a better solution

8 Optimisation 2 if labeling() giving Cost 0,Sol 0 = false then return false for i= to do pose CostVar < Cost i- if labeling() giving Cost i,sol i = false then return Cost i-,sol i- NOTE: optimality guaranteed. Last iteration gives proof of optimality. Restart from scratch each time it finds a better solution

9 Optimisation 2 if labeling() giving Cost 0,Sol 0 = false then return false for i= to do pose CostVar < 0.95 Cost i- if labeling() giving Cost i,sol i = false then return Cost i-,sol i- NOTE: within 5% of optimality guaranteed. Last iteration gives proof of optimality. Restart from scratch each time it finds a better solution

10 Optimisation 3 if labeling() giving Cost 0,Sol 0 = false then return false for i= to do if min(costvar) = Cost i- then return Cost i-,sol i- if (pose CostVar min(costvar)+ Cost i- )/2 labeling() giving Cost i,sol i )= false then pose CostVar > min(costvar)+ Cost i- )/2 Cost i,sol i = Cost i-,sol i- I.e., bisection search for the optimal cost. Can be much more efficient than Optimisation & 2. Why? Restart from scratch each time it finds a better solution

11 Optimisation 4 labeling([minimize(costvar)], DVars) Does not restart each time it finds a better solution

12 Warning Even if your labelling has to fix all variables, it should be deterministic as soon as you know that this is irrelevant for the cost! Typical example: disjunctive scheduling+makespan After fixing all disjunctions you just need to fix all the remaining variables to their minimum value.

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14 Enumeration phases Enumeration oriented for finding a first good solution (keep a low cost) Enumeration for proving optimality (increase the cost to fail as soon as possible)

15 Regret based enumeration Enumerate on decision variables by considering the impact of each decision on the increase of the lower bound of the cost variable A lot of optimization constraints can compute a regret matrix as a side effect of computing an estimation of the lower bound of the cost

16 Domination Often, one can find criteria for checking that a partial solution P will for sure have a cost smaller than or equal to the best cost of a family F of solutions: avoid to explore F by dynamically adding extra constraints Unlike OR where this is directly done in the algorithm you usually have to put it within the labelling. In this context one can sometimes use symmetries.

17 Domination example: Map coloring Objective: color a map with as few colors as possible. Constraints: X i X j whenever country i borders country j. Variable order: - Most constrained first Variable order: - Try all colors used so far - Try one unused color only The unused colors are symmetric wrt. the colored part.

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19 Global_cardinality_with_cost ORIGIN : [Régin99] ARGUMENTS : ctr_arguments(global_cardinality_with_costs, [VARIABLES-collection(var-dvar), VALUES-collection(val-int, noccurrence-dvar), MATRIX-collection(i-int, j-int, c-int), COST-dvar]). PURPOSE : In addition to global_cardinality, COST of an assignment is equal to the sum of the elementary costs associated to the fact that we assign the I-th variable of the VARIABLES collection to the j-th value of the VALUES collection. These elementary costs are given by the MATRIX collection. EXAMPLE : global_cardinality_with_costs([[var-3],[var-3],[var-3],[var-6]], [[val-3, noccurrence-3],[val-5, noccurrence-0], [val-6, noccurrence-]], [[i-,j-, c-4],[i-,j-2, c-],[i-,j-3, c-7],[i-2,j-, c-],[i-2,j-2, c-0], [i-2,j-3, c-8],[i-3,j-, c-3],[i-3,j-2, c-2],[i-3,j-3, c-],[i-4,j-, c-0], [i-4,j-2, c-0],[i-4,j-3, c-6]],4)

20 Global_cardinality_with_cost INITIAL GRAPH FINAL GRAPH

21 Global_cardinality_with_cost J-C Régin, Arc Consistency for Global Cardinality with Costs, Proc. CP 99, LNCS 73, Springer-Verlag, 999. A min-cost flow algorithm, successive shortest paths, computes: a solution with minimal cost LB a reduced-cost matrix Dijkstra s algorithm (single source shortest paths) computes a shortest-path matrix. Regret matrix = reduced-cost matrix + shortest-path matrix. val var when LB+regret[var,val]>max(Cost) Dually for the maximal cost.

22 Minimum_weight_alldifferent ORIGIN : [FocacciLodiMilano99] ARGUMENTS : ctr_arguments(minimum_weight_alldifferent, [VARIABLES-collection(var-dvar), MATRIX-collection(i-int, j-int, c-int), COST-dvar]). PURPOSE : All the variables of the VARIABLES collection are distinct. COST is equal to the sum of MATRIX[i,var i ] ( i VARIABLES ). EXAMPLE : minimum_weight_alldifferent([[var-2],[var-3],[var-],[var-4]], [[i-,j-, c-4],[i-,j-2, c-],[i-,j-3, c-7],[i-,j-4, c-0], [i-2,j-, c-],[i-2,j-2, c-0],[i-2,j-3, c-8],[i-2,j-4, c-2], [i-3,j-, c-3],[i-3,j-2, c-2],[i-3,j-3, c-],[i-3,j-4, c-6], [i-4,j-, c-0],[i-4,j-2, c-0],[i-4,j-3, c-6],[i-4,j-4, c-5]], 7)).

23 Minimum_weight_alldifferent VARIABLES :var 2:var vars.var = vars2.key ARC CONSTRAINT ctr_graph(minimum_weight_alldifferent, [VARIABLES], 2, [CLIQUE>>collection(vars,vars2)], [vars.var = vars2.key], INITIAL GRAPH [NTREE = 0, SUM_WEIGHT_ARC(MATRIX[(vars.key)*size(VARIABLES)+vars.var)]=COST). CLIQUE

24 Minimum_weight_alldifferent VARIABLES INITIAL GRAPH CLIQUE minimum_weight_alldifferent([[var-2],[var-3],[var-],[var-4]], [[i-,j-, c-4],[i-,j-2, c-],[i-,j-3, c-7],[i-,j-4, c-0], [i-2,j-, c-],[i-2,j-2, c-0],[i-2,j-3, c-8],[i-2,j-4, c-2], [i-3,j-, c-3],[i-3,j-2, c-2],[i-3,j-3, c-],[i-3,j-4, c-6], [i-4,j-, c-0],[i-4,j-2, c-0],[i-4,j-3, c-6],[i-4,j-4, c-5]], 7)). NARC=4 SUM_WEIGHT_ARC=+8+3+5=7 FINAL GRAPH

25 Minimum_weight_alldifferent M Sellmann, An Arc Consistency Alg. for the Minimum Weight All Different Constraint, Proc. CP 02, LNCS 2470, Springer-Verlag, A min-cost flow algorithm, the Hungarian Algorithm, computes: a solution with minimal cost LB a reduced-cost matrix Dijkstra s algorithm (single source shortest paths) computes a shortest-path matrix. Regret matrix = reduced-cost matrix + shortest-path matrix. val var when LB+regret[var,val]>max(Cost) Dually for the maximal cost.

26 { } 2, 7 6 3, 2 5,, 6,, sum_of_weights_of_distinct_values weight val weight val weight val var var var Sum_of_weights_of_distinct_values

27 Parameters Decision variables val sum_of_weights_of_distinct_values val 2 val 6 { var, var 6, var }, weight 5, weight 3,, 2 weight 7 Weights of values Cost variable

28 Cost Modelling Modeling the cost for problems where: using a resource once or several times costs the same (for instance cost for opening facilities in the warehouse location problem) CostW=CostW+CostW3 C C2 C3 C4 C5 customers warehouses 2 3 closed facility { var Customer, var Customer,, var Customer } 2 val weight CostWarehouse, swdv val 2 weight CostWarehouse2,,... val m weight CostWarehousem n, CostWarehouses

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30 The Different Algorithms (generic structure) swdv(variables, Values, Cost) FILTERING ALGORITHMS find BOUNDS for Cost Lower bound LB: domination Upper bound UB: matching PROPAGATE from bounds of Cost to Variables Remove val from var iff: LB+lower_regret(var,val)>max(Cost) Remove val from var iff: UB-upper_regret(var,val)<min(Cost) Propagate from min(cost) and max(cost)

31 Focus on Lower Bound swdv(variables, Values, Cost) FILTERING ALGORITHMS find BOUNDS for Cost Lower bound LB: domination Upper bound UB: matching PROPAGATE from bounds of Cost to Variables Remove val from var iff: LB+lower_regret(var,val)>max(Cost) Remove val from var iff: UB-upper_regret(var,val)<min(Cost) Propagate from min(cost) and max(cost)

32 Evaluating a Lower Bound of Cost MAIN RESULT An O(n logn+m) algorithm which gives a tight bound when all domain consists of only one interval of consecutive values, where: n m is the number of variables, is the number of values.

33 Evaluating a Lower Bound of Cost IDEA : select a subset of variables Var of V,V2,,Vn such that If the domain of every variable is an interval, then any set of values covering all variables of Var allows also to cover all variables of V,V2,,Vn.

34 Evaluating a Lower Bound of Cost IDEA 2: If dom(vi) is included in or equal to dom(vj) then the sum of the weights of the distinct values does not change if Vj is not considered

35 Evaluating a Lower Bound of Cost IDEA 3: Assume that, for each possible value v of, we know a tight v lower bound tlb,2,, p for covering all variables V, V2,, according to the hypothesis that we use value v. V p+ Let be a variable such that : ( ) ( ) ( dom( V ) dom( V ) ( V ) = If w dom V p then tlb,2,, p+ = tlb,2,, p, If w dom V p then tlb,2,, p+ min tlb,2,, p + weight w. p + p p i= dom w w w = v dom i ( ) ( v ) ( ) V p V p V p

36 Evaluating a Lower Bound of Cost STEP : compute a relevant subset of variables such that can use IDEA 3: p ( dom( V ) dom( V ) ( V ) = + p p i= dom i V, V2,, V p STEP 2: compute a lower bound for that subset

37 Computing a Relevant Subset of Variables () sorting the variables on their minimum (2) making a first selection of variables (3) restricting the previous selection

38 () Sorting the Variables on their Minimum V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e

39 (2) Making a First Selection of Variables V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e IDEA 2 V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e

40 (3) Restricting the Previous Selection V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e IDEA 2 V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e

41 (3) Restricting the Previous Selection V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e IDEA 2 V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e

42 (3) Restricting the Previous Selection V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e IDEA 2 V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e

43 (3) Restricting the Previous Selection By construction the minimum values, maximum values, of the selected variables are sorted in increasing order. So that the following holds: ( dom( V ) dom( V ) p i= dom p+ ( V ) = i p V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e V V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 V a V b V c V d V e

44 Evaluating a Lower Bound of Cost (milestone) STEP : compute a relevant subset of variables STEP 2: compute a lower bound for that subset (done) (to do next)

45 Computing a Lower Bound for the Selected Variables Use formula of idea 3: ( ) ( ) If w dom V p then tlb,2,, p+ = tlb,2,, p, w w If w dom V p then tlb,2,, = min tlb,2,, p + weight w. w p+ v dom ( ) ( v ) ( ) V p DIFFICULTY: efficient evaluation of min v dom ( ) ( v tlb ),2,, p V p Relaxed to: min ( ) ( ) ( v min tlb ),2,, p V v max V p p

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47 Cost Filtering with Side Constraint: Context Lower bound for Cost: lb Regret matrix: A i = val j Cost lb+regret ij CTR ({A,A 2,...,A n }, Cost ) A val val 2... val m A 2... regret i j A n CTR 2 ({A,A 2,...,A n }) (graph property) Polynomial algorithm for checking a necessary condition: check_ctr 2 ({A,A 2,...,A n }): (maybe,no) How to get a better evaluation of min(cost) according to CTR 2?

48 Cost Filtering with Side Constraint: Method Bisection search for the smallest regret such that check_ctr 2 ({A,A 2,...,A n }) = maybe where the domains are restricted so that all regrets are. A val val 2... val m A i val j when lb+max(,regret ij ) > max(cost) A 2... A n Avoid shaving!

49 Cost Filtering with Side Constraint: Exercise Model the Traveling Salesman Problem with a Minimum Weight All Different + a side constraint! Design a search procedure!

50 Incremental Algorithms for Cost Filtering Assume check_ctr 2 ({A,A 2,...,A n }) checks if no more than one strongly connected component in the following graph: A :: 2 A 2 ::,3 A 3 ::,4 A 4 :: 3 A A 2 A 3 A Assignment variables Regret matrix Initial graph Efficient algorithm?